So, I have migrated Firebase on my project from v@5 to v@14 because push-notifications did not work on android API31+(v12) at all.
NOTE: we use react-native-splash-screen. Folk pointed out on some problems react-native-firebase/issues/4005 and react-native-firebase/issues/2768 and react-native-splash-screen/issues/289 and suggested to move from older splash approach to react-native-bootsplash as an alternatiove, which was not really the solution for us.
PS. any comments inside links above did not help
firebase.messaging().onNotificationOpenedApp - Unlike initial pushes, in background mode push service was performing well as expected.